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%{\LARGE\bf 上海立信会计金融学院期终考试卷 --- 试题纸} \hspace{0.3cm} {\Large \underline{ A }卷 }
{\Large\bf \H 上海立信会计金融学院期终考试卷 } \hspace{0.3cm} {\Large \underline{ B }卷 }

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{\large \bf \H 2023 $\sim$ 2024 学年 第 二 学期 }

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{\large \bf \H \underline{ \emph{2022级跨学科跨专业选修课班级} } 《\underline{ \emph{复变函数} }》 课程代码：\underline{ 162250220 }  }

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{\H（本场考试属\underline{  开  }卷考试，考试时间共\underline{  90  }分钟，不准使用计算器）共\underline{  4  }页 }
%{\large （本场考试属闭卷考试，考试时间 90 分钟，禁止使用计算器） }

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%{\large 本考试卷共 4 页，请在本考试卷上答题。}

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班级 \underline{\hspace{3.5cm}} 学号 \underline{\hspace{3.5cm}} 姓名 \underline{\hspace{3.5cm}} 

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题号 &一&二&三&四&五&六&七&八&九&十&总分&合成人签名&审核人签名 \\
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%应得分&15&15&15&15&15&15&10&100 \\
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得分 $\,\,\,\,\,\,\,\,$ &&&&&&&&&&&&& \\
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本次考试共10题，每题10分。
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\begin{enumerate}

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\item %1
函数 $w=\frac{1}{z}$ 将 $z$ 平面上的单位圆周 $x^2+y^2=1$ 变成 $w$ 平面上的什么曲线？

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\item %2
求 $\sin(5i)$ 和 $\cos(5i)$ 的值，并验证 $\sin^2(5i) + \cos^2(5i) =1$. 

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\item %3
考虑初等多值函数 $f(z)=\mathrm{Ln}(z)$. 
计算 $f(5+12i)$ 的所有值。

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\item %4
设解析函数 $f(z)=\sqrt[4]{z}$ 定义在 $\mathbb{C}-(-\infty,0]$ 上，并且 $f(1)=-1$, 试求 $f(1-i)$ 的值。

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\item %5
求出 $f(z)=z\cos(z)$ 的原函数，并使用原函数计算积分 
$$\int_{-\pi i}^{\pi i} z\cos (z) dz. $$

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\item %6
设 $C$ 为单位圆周 $|z|=1$. 使用解析函数的导数的柯西积分公式，计算 $$\int_C \frac{dz}{z^2(z^2+4)}.$$

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\item %7
求幂级数 $\sum\limits_{n=1}^{\infty} \cos(in)z^n$ 的收敛半径。

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\item %8
将函数 $f(z)=\int_0^z \frac{\sin z}{z} dz$ 展成 $z$ 的幂级数，并指出展式成立的范围。

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\item %9
判断函数 $f(z)=xy^2+ix^2y$ 的可微性与解析性。


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\item %10
设 $C$ 为圆周 $|z|=1$, 分别使用柯西积分公式和柯西积分定理，计算积分 
$$
\int_C \frac{\sin{z}}{z}dz. 
$$



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\end{enumerate}
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